Hubble’s Law and the Expanding Universe A. Hubble’s Law Introduction Normally when we want to know how fast an object is going, we watch how long it takes the object to travel from one position to the next. Since it takes a long time for galaxies to move a significant distance, we can’t use this method. Instead of comparing galaxy positions measured at two different times, we measure the velocities of the galaxies (relative to our home galaxy, the Milky Way) directly. To measure a galaxy’s velocity, we make use of a phenomenon called the Doppler shift. When a source of light waves is traveling away from you, the wavelength of the light waves will be slightly longer than the wavelength you would observe if the source were stationary. In astronomy, this shift toward longer wavelengths is referred to as a “redshift” since red is at the long-wavelength end of the visible spectrum. Conversely, a shift toward shorter wavelength, which occurs when the source is moving toward you, is called a “blueshift.” Since the light from galaxies comes mostly from stars, galaxies have absorption lines in their spectra, just as stars do. The lines represent a sort of average of the spectra of all the stars contained in the galaxy. As we learned in the spectroscopy lab, these spectral lines occur at very specific wavelengths, determined by the elements in the stars’ atmospheres. The position of a spectral line as observed in the laboratory, when the source is at rest relative to the observer, is known as its “rest” wavelength. When a galaxy is moving away (receding) from us, the spectral lines are shifted toward longer wavelengths, i.e., toward the red, for lines in the visible part of the spectrum. The amount of the shift is directly proportional to the velocity at which the galaxy is moving. Thus by measuring how a spectral line has shifted from its “rest” wavelength, we can measure the galaxy’s velocity. I. Measuring galaxy velocities using the Doppler shift of spectral lines. Here you will measure the redshift of the spectral lines of five different galaxies. From the amount of the redshift you will deduce how fast the galaxies are moving relative to the Milky Way. Take a look at the five galaxies and accompanying spectra on the next page. The horizontal white band in the middle of each spectrum is the absorption spectrum of the galaxy, with an emission spectrum for comparison above and below the galaxy’s spectrum. These spectra are in black and white, but notice on the scale at the bottom that wavelength increases to the right. Longer wavelengths correspond to a redder color, so the right edge of each spectrum is the red end, and the left edge is the blue end. Next, notice the pair of dark vertical lines in each galaxy’s absorption spectrum. These lines are due to calcium in the atmospheres of stars in these galaxies, and are known as the calcium “H and K” lines. We will use the H line (the one on the right) to measure all the redshifts. The emission line spectra above and below each galaxy spectrum are for reference. The line of interest to us is the calcium H line, at 396.5 nm. It’s the second emission line from the left in the reference spectra.

Step 1: Use a ruler to draw a vertical line through all five spectra, at the position of the comparison line at 396.5 nm. Take care to draw it as accurately as possible, so that it goes through the center of the comparison line in each reference spectrum. Step 2: Use a ruler to measure the horizontal distance between the position of the comparison line and the position of the calcium H line. Measure the separation in millimeters, estimating to the nearest tenth of a millimeter, and record your results in column 1 of Table 1 for each galaxy. To determine the redshifts in nanometers that correspond to the offsets measured in millimeters in Step 2, we need to know the scale of these spectrum images. The scale tells you how many nanometers each millimeter of shift corresponds to. It allows you to convert a measurement in millimeters to a measurement in nanometers, which is what we need for a redshift. Measure the scale as follows: Step 3: Compute the separation in nanometers between the reference lines at 388.8 nm and at 501.5 nm (subtract one from the other). Record your result below Table 1. Step 4: Use a ruler to measure the horizontal separation in millimeters between the centers of these two reference lines. Record your result below Table 1. Step 5: Divide the separation in nanometers by the separation in millimeters. The result will be the scale, in nanometers per millimeter. This tells you how many nanometers of the spectrum there are in each millimeter on the page. Record your result below Table 1. Now you can compute the redshift corresponding to each of the shifts you measured in Step 2 above, as follows: Step 6: Multiply the shift in millimeters (from column 1 of Table 1) by the scale in nanometers per millimeter to determine the redshift in nanometers of each of the galaxies. Record your results in column 2 of Table 1. To determine how fast each of the galaxies is moving, we now need to know the relation between redshift and velocity. The Doppler formula is: v = (Δλ/λ) × c where v is the velocity of the galaxy, Δλ is the redshift in nanometers, λ is the rest wavelength of the line being measured, and c is the speed of light in km/s. Thus, Velocity of Galaxy = (redshift / 396.5 nm) × 300,000 km/s Step 7: Use the redshifts from Table 1 and the formula above to compute the velocity of each galaxy. Record your results in column 3 of Table 1. Q1: Look at the velocity column in Table 1. How do the velocities of these galaxies compare with velocities commonly observed on the surface of the Earth? (Hint: 1 km/s is about 2,200 miles per hour.)

Q2: The Andromeda galaxy, M31, which is one of the nearest galaxies, has a blueshifted spectrum. What can you deduce about its motion relative to the Milky Way? II. The Hubble Constant and the Age of the Universe In 1929, Edwin Hubble examined a series of velocity measurements similar to those you made in Exercise A above. The distances to the galaxies had also been measured, using Cepheid variable stars in the galaxies. He noticed that the farther away the galaxy, the faster it moved, and that the relationship was linear: a galaxy twice as far away as another will move twice as fast, and a galaxy five times as far away will move five times as fast. This relationship between the velocity of a galaxy and its distance has come to be known as the “Hubble Law.” It provided the first evidence that the Universe is expanding, a result that was rather startling at the time, as it had long been assumed that the Universe was static and unchanging. In this exercise, we will measure the Hubble Law, and then see how it can be used to determine the distances to galaxies and the age of the Universe. (a) The Hubble Law Table 1 (from Ex. A) contains all the information needed to measure the Hubble Law: the velocity of each galaxy, which you computed in Ex. A, and the distance to each galaxy, which has been provided (and could be obtained, for example, using Cepheid variable stars). Notice that the velocities are all given in kilometers per second (km/s) and the distances in megaparsecs (Mpc). 1 Mpc is a very, very long distance, which is why astronomers use it when comparing the distances to galaxies.

1 megaparsec = 1,000,000 parsec = 3,262,000 light-years = 3 x 1019 km Step 1: On a sheet of graph paper, plot the galaxies’ distances (in Mpc) on the x-axis, and the galaxies’ velocities (in km/s) on the y-axis. Make sure your graph uses most of the page and label the units along your axes. Your x-axis should go from zero to 1,200 Mpc and your y-axis should go from zero to 80,000 km/s. Step 2: Using a ruler, estimate by eye the straight line that comes closest to going through all five points (“best fit line”) and the origin. Begin the line at the point (0,0), which represents our home galaxy, the Milky Way (zero distance and zero velocity). Since the measurements are inherently imperfect, and galaxies can also have velocities that deviate somewhat from the Hubble Law, the galaxies won’t all lie exactly on the line. Draw in the line that you think comes closest. The line you have drawn on your graph represents your best determination of the Hubble Law. This law can be used to determine the distances to other galaxies. The one exception is that it can’t be used for very nearby galaxies, such as those in the “Local Group” near the Milky Way. For these galaxies, random velocities toward or away from the Milky Way can be larger than the expansion velocity, so the Hubble Law doesn’t give good distances. In fact, some nearby galaxies actually move toward us, rather than away (for example, M31). For more distant galaxies, however, the expansion velocity is very large, so that random velocities are negligible by comparison. For these galaxies, the Hubble Law provides a good means of estimating distance.

To estimate the distance to a galaxy of known velocity, locate that velocity on the vertical axis on your graph, and then move across the graph until you hit the Hubble Law line. Then move down vertically to see what distance this corresponds to. Use your graph with the best-fit line to answer the next two questions. Q3: What is the distance to a galaxy whose velocity is 50,000 km/s? Don’t forget units! Q4: What is the distance to a galaxy whose velocity is 10,000 km/s? Don’t forget units! Step 3: Distances can also be determined numerically, if we first determine the slope of the Hubble Law. This slope is generally referred to as the “Hubble constant,” To measure the Hubble constant, extend the line you drew in Step 2 to the edge of the graph paper. Q5: Use your best-fit line to find the velocity of a galaxy that is located at a distance of 1000 Mpc. Step 4: The Hubble constant (often called just “H”) is the ratio of these two quantities (distance and velocity), expressed in km/s per Mpc. Q6: Using the results from Q5, divide the velocity by the distance to determine your value of the Hubble constant in units of km/s/Mpc. You can think of the Hubble constant as the velocity (in km/s) of a galaxy at a distance of 1 Mpc. Since the Hubble Law is a linear relation, the velocity of any other galaxy can then be determined by multiplying its distance in Mpc by the Hubble constant:

V = D x H (velocity = distance times Hubble constant) Alternatively, you can determine the distance to a galaxy using

D = V / H (distance = velocity divided by Hubble constant) In these formulae, the distance is always in Mpc, and the velocity is always in km/s. Q7: Using the formula D = V / H, compute the distance to a galaxy whose velocity is 20,000 km/s. (b) The Age of the Universe One way to estimate the age of the Universe is to determine how long it has taken various remote galaxies to move to where they are now from where they were at the time of the Big Bang, when the Universe began expanding. If you know the velocity of a galaxy and the distance it has traveled, you can determine the time it has taken that galaxy to move to where it is now (assuming its velocity has remained constant). The distance a galaxy has traveled is equivalent to its current distance from the Milky Way, because in the early Universe, all matter was very densely packed together.

We know that in general, distance = velocity x time, or D = V x T For example, if you travel at 60 miles per hour for 2 hours, you will have traveled 120 miles. This relation can also be written as T = D / V. In our example, this means the time of travel is equal to 120 miles divided by 60 miles per hour, or 2 hours. This formula can be used to determine the age of the Universe, given the distance and velocity of a galaxy (or preferably, of many galaxies). Notice that the Hubble constant is in the form of a velocity divided by a distance:

H = V / D Thus the inverse of the Hubble constant is

1 / H = D / V But this is the same as the expression for time of travel above. In this case it represents the time of travel of galaxies since the Big Bang, which is to say, the age of the Universe. Q8: Compute the age of the Universe, using the value of H you found in Q6. Divide 1 by the value of H to get the age of the universe in units of Mpc/(km/s). These are very awkward units in which to express a time! To convert to an age in years, recall that 1 Mpc = 3 x 1019 km. We’ll also use the fact that 1 year = 3 x 107 s, or 1 s = 1/(3x107) yr. So we have 1 Mpc/(km/s) = 3 x 1019 km / (km/s) = 3 x 1019 s = 3 x 1019 / 3 x 107 = 1012 yr Q9: Multiply your answer from Q8 by 1012 to convert your answer from Mpc/(km/s) to years. Next, let’s check whether all galaxies have taken the same amount of time to get to where they are. Q10: Use the relation T = D / V to determine the time it has taken each of the galaxies in Q3, Q4, and Q7 above to get to where they are now. Use the conversion factor above to convert your answers to years in all cases. Show all your work. Q11: What do you notice about the times for each of the galaxies? Why do you think this is so? B. The Expanding Universe (Adapted from the Project ASTRO Resource Notebook) I. Visualizing the Expansion of Space The last two sheets of this lab include a sheet of paper containing many dots, labeled ‘The Universe Today’, which corresponds to your transparency sheet, and another sheet of paper

containing a similar (but not identical) distribution of dots, labeled ‘The Universe One Billion Years Ago’. Each dot represents a galaxy in a model 2-dimensional Universe. We will compare the dots on the paper representing the Universe 1 billion years ago and the dots on the transparency representing the Universe today. You can think of them as two snapshots in time of a small section of the Universe. Q12: Take a close look at the dots on the sheet of paper labeled “The Universe One Billion Years Ago”. Is there any center to the distribution of dots, or does the distribution appear to be relatively uniform? Q3: Now take a close look at the transparency by itself. Does this distribution have a noticeable center? Now place the transparency on top of the sheet of paper with the dots on it (make sure you use the “Universe One Billion Years Ago). Take care not to rotate one sheet relative to the other. Don’t worry that the outlines of the regions may not line up perfectly; just keep the lines on the two sheets parallel. Examine the pattern of dots. Q14: What does the pattern of dots look like? Is there any center to the pattern now? If so, describe the pattern. Now shift the transparency relative to the paper, again being careful not to rotate it relative to the sheet underneath. Try shifting it in different directions, up and down, left and right, keeping the edges of the boxes parallel to one another. Q15: What happens to the center point when you shift the transparency to the left? What about when you shift it to the right? Now pick a particular dot on the sheet of paper to be your home galaxy. Pick a galaxy that is not right at the center of the sheet, and is also not too close to the edge (e.g. at least a centimeter from the edge). The idea is for each person to pick a different dot as his or her home galaxy. Later we will compare results obtained from different galaxies. Q16: Draw a rectangle on your answer sheet to represent the outline on the transparency, and mark a small dot to indicate the location of your home galaxy. Next, place the transparency on top of the sheet of paper, and shift it around until your home galaxy is at the center of the pattern. Match your galaxy to its corresponding dot on the transparency. The sheet and transparency overlay are like two snapshot photos taken 1 billion years apart. The separation between a given dot on the sheet and the corresponding dot on the transparency tells you how far the galaxy has moved in 1 billion years. By placing your home galaxy at the center of the distribution, you will effectively be making all measurements of other galaxies relative to your home galaxy.

NOTE: If it’s hard to keep the transparency and paper aligned, use some masking tape to keep the transparency fixed in place on top of the sheet of paper.

Q17: From your perspective (sitting in your home galaxy), how far has your home galaxy moved in 1 billion years?

Now take a look at what the rest of the galaxies are doing, from the perspective of your home galaxy. Q18: (i) In what direction are the other galaxies moving, relative to you? (ii) Are all the galaxies moving at the same speed relative to you? How can you tell? (iii) What relation, if any, do you see between the speed of the galaxies (relative to your home galaxy) and their distances from your home galaxy? II. Calculating the Age of the Universe We can use the pattern of dots to estimate the age of our 2-dimensional model Universe. What we want to know is how long the Universe has been expanding, which is equivalent to its age, since the Universe began expanding as soon as it was formed at the time of the Big Bang. At that time, everything in the Universe was very close together. In our 2-dimensional model, all the dots on the sheets would have been so close together that they would have looked like a single dot, at the position of your home galaxy. We can use a single galaxy to estimate how long the expansion has been going on, but a better estimate will be obtained by repeating the same measurement for several galaxies. Here’s how it works. You can measure how far a particular galaxy travels in one billion years by measuring how far it has moved between the snapshot “one billion years ago” and the snapshot of “the Universe today.” You can determine how far that same galaxy has moved since the time of the Big Bang by measuring its separation from your home galaxy on the transparency (Universe today). Once you know how far it moves in 1 billion years, and how far it has traveled altogether, you can work out how many billion years it took to travel the whole distance. Q19: Let’s do this process for six galaxies. Pick six that are different distances from your home galaxy (e.g., 1-2 cm, 2-3 cm, 3-4 cm, 4-6 cm, 6-8 cm, and 8-10 cm away from your home galaxy). For each galaxy, measure the following two quantities, and record them in Table 2 on the worksheet.

(i) The total distance it has moved from the home galaxy

(ii) The small distance it has moved in 1 billion years (i.e. the distance between the dot on the sheet and the dot on the transparency that represent the same galaxy) To determine the age of the Universe from each individual galaxy, DIVIDE (i) by (ii). Your best estimate of the age of the Universe can now be obtained by taking the mean, or average of all the individual measurements. The average is found by adding the results of the six measurements (add up the numbers in the last column), and then dividing the sum by the number of measurements (six). Q20: Calculate the average age of the universe from the data in Table 2, following the steps below.

In order to be able to make meaningful comparisons between the age you determined and the ages other students found, it’s essential to get an idea of how accurate your measurement is. No measurement is perfect. But how good is it? A widely used method to estimate the uncertainty in any set of measurements of the same quantity is to determine what is known as the “standard deviation,” and is represented by the Greek letter “sigma” (σ).

xi is an individual measurement x is the average of all measurements N is the number of measurements Σ means “sum over all measurements” The steps in determining the standard deviation are as follows. Table 3 on your worksheet will lead you through each of these steps. Fill it out as you go!

(i) Find the difference between each individual measurement and the average of all the measurements (subtract average age from individual age) (ii) Square each of the differences (i.e. multiply the difference by itself) (iii) Add together the squares of all the differences (iv) Divide the sum by one less than the number of measurements (v) Take the square root.

Q21: Express your final result as the average age plus or minus the standard deviation. About 68% of all measurements of the same quantity should agree with each other to within one standard deviation. About 95% of measurements will agree within two times the standard deviation. As a rough rule of thumb, variations in the measurements should be larger than about 3 times the standard deviation to be considered evidence of real, underlying variations in the quantity being measured. Smaller deviations are likely to be due to measurement error. Write your results on the whiteboard in the format of “XX +/- Y billion years”. We’ll compare results for different groups, who chose different galaxies as their “home” galaxy. Q22: How does your computed age for the Universe compare with the ages found by other groups who used different centers for their home galaxy? Q23: How might you account for any differences between your answers and those of your classmates? Q24: Do you think astronomers in distant galaxies would agree with astronomers on Earth about the age of the Universe? Why or why not?

( )1-Nx -x

2i!="

Hubble Law Worksheet

Separation = _____________ nanometers (nm)

Separation = _____________ millimeters (mm)

Scale = ______________ nm per mm

The Expanding Universe Worksheet

Table 2: Average Age of the Universe

distance from home galaxy today ÷distance traveled in 1 billion yr = age of universe 1. ______________________ ÷ ________________________ = ______________

2. ______________________ ÷ ________________________ = ______________

3. ______________________ ÷ ________________________ = ______________

4. ______________________ ÷ ________________________ = ______________

5. ______________________ ÷ ________________________ = ______________

6. ______________________ ÷ ________________________ = ______________ Table 3: Standard Deviation

age average age difference difference squared

1. _________ – __________ = ________ ____________

2. _________ – __________ = ________ ____________

3. _________ – __________ = ________ ____________

4. _________ – __________ = ________ ____________

5. _________ – __________ = ________ ____________

6. _________ – __________ = ________ ____________

Next, add up the numbers in last column of Table 3 (“differences squared”): Then, divide this by the number of measurements minus one (i.e., divide by 5): Finally, take square root: